Analytic approximations for the Fermi energy of an ideal Fermi gas

An important function in semiconductor-device analysis and transport theory is the widely tabulated Fermi-Dirac integral, ℱ (η) =2π−1/2ℱ∞0[exp(x−η)+1]−1f dx, f=x1/2, which relates, for example, the Fermi energy ηkT to the carrier density N=ℱN0 in a parabolic semiconductor band (N0=effective density of states). We show that the classical or Boltzmann approximation to this integral (η=lnℱ, η≲−2) is extended to cover the Fermi-energy range of semiconductor lasers (η≲+2) by the expression η=lnℱ+2−3/2ℱ and by other simple differentiable approximations applicable to higher degeneracy (η≲7) or to nonparabolic bands (f≠x1/2).